Spurious correlations, three ways

I stumbled across a collection of spurious correlations a while back.

Here is one of them that struck me, in part because the data set links sour cream consumption per capita and motorcycle riders killed in non-collision transport accidents, but also because it is presented in odd sort of way.

Headline: Sour cream consumption linked to greater risk of non-collision transport death by motorcycle.

Why Precision Matters

The following home workshop was used with preservice teachers in an effort to highlight the importance of attending to precision when doing mathematics investigations.

Home Workshop - Angles of Polygons by hasenbaj

Phronesis? (Using Tools Strategically)

I was preparing a lesson on representing data with dot plots and box plots. The goal was to help students recognize the pros & cons of each type of visual display, among other things (like simply making sure everyone knew how to construct one).

But it turned into something more powerful, at least for a moment, and I thought I would share that moment with you via this post.

The lesson also built off the previous session where we had used "MAD Minute" type tasks to generate data. You know, like these:

 

Lots of kids are good at those. I was good at them. Many of my preservice elementary teacher students said they loved them too. "Oh man, I loved doing these!" But you didn't love them if you weren't good at them. And we had that discussion: For some kids, these high-stakes timed worksheets were sources of extreme anxiety.

What (else) it means to solve

In class yesterday, I had an opportunity to reinforce what it means to solve equations and inequalities (a connection I've discussed in a previous post). My students and I were working on a task that required that we verify that x = 2 and x = 0 are the two solutions of the quadratic equation -2x^2 + 4x + 1 = 1. How can we do that? A chorus of replies: "plug them in and check." 

"Right," I said, "We need to see if we have found both (all) of the values that make the equation true."

Learner #1 raised his hand and asked, "Doesn't it also mean finding the points where the parabola intersects the line y = 1?" I know from a previous conversation (described in this previous post) that this was how he learned to solve quadratics in high school: enter the parabola into Y1, enter the other expression into Y2, and use the CALC -> INTERSECT to solve.

Created with Geogebra

So I was happy to affirm the connection.

A short time later, in the same lesson, we were exploring the function f(x) = (x-3)^2 + 5. We had determined that because the parabola opened upward and had its vertex at (3,5), so there must not be any solutions to the quadratic equation f(x) = 1. Furthermore, we agreed, the equation f(x) = 5 would have just one solution (at the vertex) and f(x) = k for any k > 5 would have two real solutions.

Created with Geogebra

Then Learner #2 offered: "So if we ever find that there's no solution, we can conclude that the curves do not touch, right?" It was a nice extension of Learner #1's remark.

But it depends a bit on what you mean by "no solution," doesn't it? The situation is exactly analogous to the 1st grader who asks the question, "Teacher, what's 1 minus 3?" to which the teacher response helpfully, "Honey, you can't do 1 minus 3." I like to imagine a precocious 1st grader reply: "Yeah, maybe you can't, but I can! It's negative 2!"

What it means to solve

"Do we understand" was on my mind as I was grading intermediate algebra quizzes over lunch today. In particular, I wondered: do my students really understand what it means to solve an equation or inequality? It was clear enough that they knew how to do it, but do they know what they are accomplishing in the process?

One of the learning targets I have identified for my students reads:

I can explain what it means to solve equations, inequalities, and systems, and I can use this knowledge to check my answers for reasonableness and correctness. 

I'll be honest: I really love that learning target. It was inspired based on one of the Common Core State Standards for Mathematics, specifically, one of the Grade 6 Expressions and Equations standards:

CCSS-6.EE.5 [Students will] understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? ... (emphasis added)

That concept is fundamental to the work we do in algebra. Consider the wide array of situations where the sole directive is "solve". Come to think of it, one of my old doctoral advisors once described "solve" as one of the four universal directives of algebra homework sets. (Can you come up with the other three? Answers appear at the end of the post.)

In the case of "solve", the meaning is universal and boils down to "find the set of [thing(s)] that make this [thing] true". Consider: 

• "Solve 3x+4=7" means "Find the values of x that make the equation true."
• "Solve {3x + y = 4, x - 2y = -1}" means "find the ordered pairs that simultaneously make both equations true."
• "Solve y'(x) - y(x) = 0" means "find the family of functions that make the differential equation true."
• "Solve Ax = λx" means "find the real numbers λ and corresponding vectors x, called the eigenvalues and eigenvectors of A, that make the matrix equation true."
• "Solve Ax = 0" means "describe the set of vectors x, called the null space of A, that make the matrix equation true."